Search this site

Embedded Files

Ultimate List of Large Numbers - Part 1

Ultimate List of Large Numbers - Part 1

Zero0We open our list of positive numbers with the important number 0, which I consider to be the second most important number in existence, behind the number 1. Although zero is neither positive nor negative, it still serves as a natural starting point for a list like this. If you want to learn about notable negative numbers and miscellany look here.Zero's importance stems from being the additive identity, meaning that adding 0 to a number gives the original number. In fact, in formal mathematics 0 is often defined as the number x such that any number a plus x is still a. Therefore it's an inherently special number in all of mathematics and everywhere, which is also why 0 gets a range of numbers on this list all to itself.While all other numbers represent something, 0 represents nothing, making it special. But 0 can represent more than nothingness. It can also represent the very beginning or before the first, or a placeholder digit in numbers such as 50630005 with 0s in it. Additionally, when accounting for negative numbers, zero represents pure neutrality or stability—a lack of either extreme.Zero also very often leads to degenerate cases in functions, i.e. calculations that do not result in large values. For example, x+0 = x, x*0 = 0, x⁰ = 1, x^^...^^0 with as many ^s as we want is 1, and f₀(x) is equal to x+1 in the fast-growing hierarchy.However, since 0 does not normally represent a quantity of something, it took a very long time for zero to be recognized as a number like 1, 2, 3, etc. were thought of as numbers (see my large number timeline for more). The earliest known record of zero as a number was around 250 BC, and in Europe this happened around the 1500s. Before then it was only recognized as a digit, and even that took a while, largely because many numeral systems like Egyptian and Roman numerals didn't need zero.Because of zero's unique history, the name "zero" cannot be traced back to Indo-European roots like 1, 2, 3, and so on can be; rather, it's traced back to Arabic "sifr". That became "zefiro" in Italian, and was later shortened to "zero". For the name of zero, English is actually a special case among Germanic languages (like English, German, and Dutch): in most Germanic languages the word for "zero" resembles "null", and that's also true for Slavic languages such as Russian. However, in other Indo-European languages (such as Romance languages like French, Spanish, and Italian), the word for zero does resemble "zero".For an interesting bit of trivia relating to zero, see the entry for 10¹⁰¹⁰ in part 2, a number equal to one followed by ten billion zeros.Reciprocal of BIG FOOT1/FOOT¹⁰(10¹⁰⁰)This is the reciprocal of BIG FOOT, currently honored as the largest named number by the googology community. This number is therefore an insanely small number, unfathomably close to 0 - and yet there's infinitely many numbers between this number and 0, and infinitely many numbers between this and the next number or the reciprocal of any really big number.Reciprocal of meameamealokkapoowa oompa1/{LLLL........LLLL, 10}10,10 with a meameamealokkapoowa-sized array of L’sThe reciprocal of Jonathan Bowers' largest named number, called meameamealokkapoowa oompa. Since meameamealokkapoowa oompa is a ridiculously huge power of 10, its reciprocal's decimal expansion is a bunch of zeros and then 1. The number of 0's here might seem to be a lot smaller than meameamealokkapoowa oompa, but on a scale with numbers this large, it’s very far from any meaningful difference because with this number we are FAR past exponentiation.Reciprocal of a beengulus1/{10,100(0,0,4)2}Just pulling out the reciprocal of a random Bowersism. The reciprocal of this number is a superdimensional array of 100(4*1002) entries, much bigger than the array used to represent a gongulus.Reciprocal of a tristo-threagol1/E100###100###100#100#100Now pulling out the reciprocal a random Saibianism, which is approximately {10,101,2,1,3} using Bowers' notation. This number is, once again, 0.0000.....0001 with a shit load of 0s, like the previous two numbers. The number of 0s is still almost exactly the same as a tristo-threagol, relatively speaking. This is approximable as a 5-entry array, so it’s in the low mid-level googolisms.Reciprocal of Graham's number1/G(64)Because if I don't mention it, someone else will.Googolminex10^(-10^100)Also known as the reciprocal of a googolplex. This is 0.000....0001 with a googol minus one zeros after the decimal point. There’s no way we could write all those zeros out, even if we put a 0 on every atom in the observable universe.The name "googolminex" was coined by Conway and Guy in their Book of Numbers. They said that since googolplex = 10^googol, x-plex can be thought of as 10ⁿ (though it could have been anyone who first came up with this), and by analogy (plus:plex::minus:?) x-minex should be 10^-n, aka 1/10ⁿ.Interestingly, googolminex, despite being less than 1, can be considered a googolism due to its unique googological name, although Sbiis Saibian suggests that a name for a small number be called a "micronym" - he has named a few "micronyms" (example). See 21 and grangolplex/googolcentiplex for further discussion on the term "googolism".Googol-minutia-speck10^-11⁰This is currently Sbiis Saibian's smallest googolism. It is formed by combining the name "googol" with the suffix -minutia meaning the reciprocal of x with the suffix -speck which divides a number by ten billion. Sbiis Saibian suggests the term "micronym" for googolisms for numbers less than 1.To get an idea of how small this number is, imagine the volume of the observable universe, 3.5*10⁸⁰ m³. Then imagine a 10^110th of the observable universe, aka a googol-minutia-speck fraction of the universe. That would be 3.5*10^-30 m³, aka a sphere 94 picometers in diameter. You may wonder, how small is 94 picometers? 94 picometers is approximately the diameter of an oxygen atom. That means that we can imagine an oxygen atom as being approximately a googol-minutia-speck portion of the universe. Now that's a pretty small number.Googol-minutia0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001This teeny tiny number is the reciprocal of a googol, called "googol-minutia" by Sbiis Saibian. It can be easily written out, and it's close to 0 but not quite inconceivably close. It can be called a googolth, or by the -minex idea it could be called "hundredminex" (see googolminex). Since a googol can be considered to be on the (very high end) universe scale, a googol can be considered to be on the fraction of the universe scale. A small biological virus would be a googolth of the volume of the observable universe.Planck time in seconds5.390*10^-44Planck units, the smallest units theoretically measurable, are home to some of the smallest numbers known to the real world, and to be meaningful as something other than the reciprocal of a large (greater than 1) number. This is the length of the Planck time in seconds - about 2*10^43 of those times fit in a second! This is the smallest positive number Robert Munafo puts on his number list.See also the Planck length and the Planck temperature.Planck length in meters1.619*10^-35This is the length of the Planck length in terms of meters. The Planck length is the amount of length light travels in a Planck time, and it's the smallest physically meaningful length. Likewise, a Planck area is a square Planck length and a Planck volume is a cubic Planck length.See also the Planck time and the Planck temperature.One millionth0.000001A millionth of something is represented with the prefix micro-, one of the more well-known SI prefixes. Its name literally means "small". Some examples of it: Microbes can be measured in micrometers. Light takes a microsecond to travel 300 meters in a vacuum, but that’s more of an example of how fast light is. For one more example, the 2011 Japanese earthquake, known for its sheer destruction, decreased the duration of Earth’s day by 2 microseconds, which much better shows how stable and large Earth is rather than how short a microsecond is.Percentages this small are usually not noted as 0.0001%, but as the easier to understand 1 ppm (part per million). The ppm unit can be used to denote, for instance, 0.00000334 as 3.34 ppm. Also used is ppb (parts per billion), the less common ppt (parts per trillion), and the still less common ppq (parts per quadrillion).One percent0.01This is 1/100, or one percent (1%). 1% is figuratively used to represent almost none, but just as 100 is typically not a giant number, 0.01 isn't always a "tiny number". And just as 100 is an easy number to imagine in your head, 1% is an easy percentage to picture: just take one part or something out, leaving 99 parts behind. It's a sizable proportion when dealing with large numbers of entities—for instance, if a country has 50 million people and 1% of them have a certain disease, then that's a respectably large 500,000 people. A 1% chance is a probability that it would often be unwise to dismiss as rare. (see 0.99)One twentieth0.05This 1/20, often referred to as 5%. 5% is easily perceivable as more than 1% but less than 10%. It is one of the more familiar percentages in life.One half0.5This 1/20, often referred to as 5%. 5% is easily perceivable as more than 1% but less than 10%. It is one of the more familiar percentages in life.One minus the reciprocal of Rayo’s number1-1/Rayo(Googol)This 1/20, often referred to as 5%. 5% is easily perceivable as more than 1% but less than 10%. It is one of the more familiar percentages in life.One1One is arguably the most important number in existence, defined as the number x such that any number a multiplied by x is still a. It has many unique properties (mostly for trivial reasons) and it also plays a unique role among the numbers.One minus the reciprocal of Rayo’s number1+1/Rayo(10¹⁰⁰)Yep, we are beginning our journey through large numbers very very slowly, with an INSANELY small large number - remember that in the entry for 1 I talked about how Sbiis Saibian proposed that a number between 0 and 1 should be considered a small number and a number between 1 and infinity should be considered a large number (1 is neither large nor small). This number is 1, a decimal point, a shit ton of zeros, and a mysterious sequence of numbers that we will never know how it starts or ends. Even if you raised this number to the power of something like Loader's number you wouldn't make it close to the next entry ...Square root of 31.732...Another very important irrational number that crops up largely in geometry. For example, the height of an equilateral triangle with side length a is equal to a*√(3)/2.Square root of Pi1.77245...This number is the square root of pi, a number that crops sometimes in mathematics. It is the area underneath a bell-curve of height 1 and standard deviation 1. It is also equal to Γ(1/2) using the gamma function, a generalization of the factorial to any real or complex inputs. Since Γ(n+1) for positive integer n = n!, this number can be thought of as -1/2!, the factorial of -1/2. Both of these are examples of pi cropping up in a place you don't expect it to.Two2Like one, two is another extremely important number, and the first integer that can only be defined from existing numbers. It also has many unique properties.Square root of 52.236...The square root of five crops up sometimes in mathematics. It isn't as common as the square roots of 2 and 3, but still important, most notable in the definition of the golden ratio.ln(10)2.302585...The square root of five crops up sometimes in mathematics. It isn't as common as the square roots of 2 and 3, but still important, most notable in the definition of the golden ratio.e2.71828...This is a well-known mathematical constant that competes with the square root of two as the second best known irrational constant, behind pi. It is also called Euler's number since this important constant was made famous by Leonhard Euler. However, he was not the first person to use the constant—that would be Jacob Bernoulli, who would be really annoyed about this if he didn't also have plenty of mathematical things named after him.e can be defined in multiple ways: the limit of (1+1/x)^x as x approaches infinity, the number a such that the derivative of a^x is itself (that means that at any point, the slope of the graph of e^x is the same as the x-coordinate of the point), the number a such that the function a^x has slope 1 at x = 0, etc. It varies greatly what people first learn e to be, and I first learned it as the number a such that the function a^x has slope 1 at x = 0.Another way to calculate e is with the sum:1 + 1/1! + 1/2! + 1/3! + 1/4! ... (sum of the reciprocals of the factorials)In fact you can calculate e^x for any x with the sum:1 + x/1! + x²/2! + x³/3! + x⁴/4! ...e appears most often in calculus when working with functions, and there it crops up again and again. It even appears a few times in googology, like when working with infinite power towers, estimating large factorials like the contrived googolbang, and in the definition of Skewes' number.Three3Three, the result of adding one to two, is another number with many unique properties. It's also known for being an inherently appealing number, with many famous occurrences.π / Pi3.14159265....Pi is the most well-known of all the irrational numbers. It's most commonly defined as the ratio of the circumference of a circle to its diameter. People knew about it since ancient times; ironically, despite ancient Greek mathematicians knowing about the number, it was not represented with the Greek letter pi until the 1700s.Pi shows up in mathematics when working with circles and other geometric functions such as the trigonometric ratios, but it also shows up in a lot of places you wouldn't expect it to (example). This is part of why pi has been given a lot of cult significance as a number - people have memorized pi to insane amounts just for the sake of it, and the current record for digits of pi memorized is 100,000! Many people have pointed out some unusual coincidences regarding the digits of pi as well, as is often the case with other cult numbers.Because pi's digits go on forever, it is a common misconception that pi is an infinite number; therefore, pi has sometimes been mistakenly thrown into the large number discussion as an example of a really large number. Well guess what: an even more incredible number is 3.2, or better yet, the unfathomably large 4! Something some might consider a little more clever would be pi with the decimal point removed, but since such a "number" is not finite, it's the same thing as throwing infinity into the large number discussion. As I said in the introduction to this site, throwing infinity into the large number discussion is cheating, and nobody likes cheaters.However, pi does indeed legitimately show up a few times in googology, most notably in Stirling's factorial approximation, a method to estimate large factorials like the googolbang. It also appears in googolisms like Ballium's number and piplex, but those don't really reflect pi's use in googology as much as its popularity as a number.Square root of 10 / Halfplex3.16227...Another less accurate approximation for pi, but a somewhat convenient one in ancient times because it's the square root of our numeral base.Square root of 123.46410...Once as a child (first or second grade), for whatever reason I thought pi was the square root of 12, until my mom told me that it wasn't. I have no idea why I ever thought such a ridiculous thing, but nonetheless I was surprised when I later learned that the square root of 10 (not 12) was sometimes used as an approximation for pi.Four4Four is a number with unique properties such as being the smallest composite, the smallest non-trivial perfect power, a common occurrence with twos in googological functions. It is also, among other things, a number things are very often grouped in.Five5Five is yet another number with zillions of properties - prime, Fibonacci, a Fermat number, part of the only twin prime triplet, the number of Platonic solids, etc. It's also significant as half of the base of our numeral system, and one of the larger numbers we can generally recognize at once.Six6Some of six's properties is that it's the smallest perfect number, a triangular number, a factorial, primorial, the number of faces on a cube, the number of trigonometric functions, etc. (READ MORE)Six is also the boundary between class 0 and class 1 numbers, and the base of Robert Munafo's entire idea of classes - class 0 numbers are numbers small enough to recognize immediately, and they range from 1 to 6. Then class 1 numbers are numbers too large to recognize immediately but still physically perceivable, and they go range from 7 to a million. Read my page on 6 (link above) for more on that.Tau / τ6.28318531... = 2*πSome of six's properties is that it's the smallest perfect number, a triangular number, a factorial, primorial, the number of faces on a cube, the number of trigonometric functions, etc. (READ MORE)Six is also the boundary between class 0 and class 1 numbers, and the base of Robert Munafo's entire idea of classes - class 0 numbers are numbers small enough to recognize immediately, and they range from 1 to 6. Then class 1 numbers are numbers too large to recognize immediately but still physically perceivable, and they go range from 7 to a million. Read my page on 6 (link above) for more on that.Seven7Seven is a prime number, a Mersenne number, a factorial prime, the smallest number of sides of a regular polygon not constructible with compass and straightedge, etc. It is a number that has a famous cult significance, given much spiritual significance since ancient times.Eight8Eight is the second and smallest non-trivial cubic number (2^3), a power of 2 (2^3), and the largest cubic Fibonacci number. It's also the number of vertices on a cube, as well as the number of cardinal directions most people would choose from when specifying on a map (north, northeast, east, southeast, south, southwest, west, northwest).8 is also the number of bits in a byte (stands for "by eight") in computing, giving it a lot of digital significance. Since a bit can store 0 or 1, a byte can store 256 different values. Bytes are also multiplied by powers of 1000 (or 1024) to give kilobytes, megabytes, gigabytes, etc. For more on that see the entry for 1024, a general discsussion of the convenience of powers of 2 to us Earthlings.Eight occurs commonly in nature, in creatures such as spiders or octopi (the name "octopus" even means "eight feet"). A set of eight also often has the sense of completeness a set of four has.The prefix octa- (both Latin and Greek, such as October, octagon, octahedron, octopus, octave) is used for eight, and iit is one of the more commonly used number prefixes.Nine9Nine is 3^2, the third square number. It is the smallest odd composite number. It is also the maximum number of cubes needed to sum up to any positive integer, as well as the third expofactorial (3^2^1, see 262,144) and the fourth subfactorial (see 44).9 and 8 are a unique pair of numbers. They are the only pair of perfect powers that neighbor: 8 = 2³ and 9 = 3². They're also the only positive integer pair of x^y and y^x, x < y, where x^y > y^x. For more on this kind of thing see the entry for 16, the only nontrivial integer case of x^y = y^x, and for more on close perfect powers see 26, the only number to fall right between a square and a cube. also, 8 and 9 add up to 17 :)9 is the largest number that is normally represented with a single glyph - beyond this point multiple digits are combined using place-value notation, which is today the dominant way to denote numbers. Place value notation has been used since the time of the Sumerians (see my large number timeline for more on that).Because 9 is the largest single-digit number, it's easy to test for divisibility by 9. Just add the digits of the number repeatedly until you end up with a single-digit number - if that number is 9 then it's divisible by 9. For example, take 76,347:7+6+3+4+7 = 272+7 = 9 - therefore 76,347 is divisible by 9. This method is known as "casting out nines", and works for the largest single-digit number in whatever base you're using.Prefixes for 9 include nona-/nov- (Latin, like in November) and ennea- (Greek, like in enneagon and enneagram). They’re relatively uncommon, but not obscure.Ten10Ten is a number that is both triangular and tetrahedral, and it also has a special property discussed at the entry for 39 (sum of prime numbers 2 to 5 and the product of 2 and 5). But its importance in humanity definitely comes from being the base of our numeral system.Booga-e (by Nayuta Ito's proposal)~ 10.30495187Up-arrow notation is a famous way to make insanely large numbers - if you're not familiar with it read here. However, a notable gap in the notation is its lack of support for arguments that are not nonnegative integers, such as 1.5, -3, or pi.To solve this problem, Nayuta Ito of Googology Wiki made a proposal to define up-arrows for any positive real arguments - for example, in that system we can define values like pi^^^e. But better yet, the system allows us to define fractional up-arrows - for example we can have 3{1.5}3 ({x} means x up-arrows), which is equal to about 144.023. The system has these three rules:b>1 and n>1: a{n}b = a{n-1}b-10≤b≤1 and n>1: a{n}b = a^b0≤b≤1 and 0≤n≤1: a{n}b = a^bn*b^1-nThis particular number is defined with Sbiis Saibian's booga- prefix, a prefix defined as booga-x = n{n-2}n. For example, booga-four = 4{4-2}4 = 4{2}4 = 4^^4, a number with about 8*10^153 digits. For more on the booga- prefix read here.With that generalization of up-arrows, we can define booga-x for any positive number x>2. This number is booga-e in that system, equal to about 2.718{0.718}2.718. The number is equal to exactly e^ee-2*e^e-3, or about 10.30495.Another number you can define with that system is booga-pi, which isn't too much bigger.Eleven11Eleven is the fifth prime number, and the smallest two-digit prime. It's the smallest repunit prime, and the next is equal to 1,111,111,111,111,111,111 - see that number's entry for more.Multiples/powers of eleven tend to be interesting: for example, 121, 1331, and 14,641, the square, cube, and fourth power of eleven, coincide with the first few rows of Pascal's triangle, a very important triangle of numbers in mathematics. Another property is that when you take a multiple of 11, reversing it gives another multiple of 11. All this stems from 11 being one more than the base of our numeral system, and is thus something specific to base 10. This kind of stuff is also reflected in the School House Rock song "Good Eleven", which talks about multiplying by eleven and how easy it is.It's also easy to test for divisibility by 11 - my favorite way is to subtract the last digit from the rest of the digits, and repeat that process - if you get 0 then it's divisible by 11, and if you don't it isn't. For example:Test 45,056:4505 - 6 = 4499449 - 9 = 44044 - 0 = 444 - 4 = 0 - therefore it's divisible by 11.This makes eleven the fourth easiest prime number to test for divisibility in base 10: the easiest are 2 and 5 (check the last digit), followed by three (sum of digits is divisible by 3). The rest of the prime numbers are harder to test for divisibility in base 10.Even better, this divisibility test also gives you the quotient when dividing a number by 11 - the quotient is each digit you subtracted, read in reverse order - in our case, the digits subtracted to test 45,056 for divisibility by 11 and 6, 9, 0, and 4, and therefore the quotient 45,056/11 = 4096.The name "eleven" literally means "one left" - like twelve it doesn't match with the pattern of names "thirteen" to "nineteen".Because things often go on a scale from one (or zero) to ten (since ten is our numeral base), "up to eleven" is a common idiom for taking something beyond the extreme - the idiom is usually credited to have originated from the movie "This is Spinal Tap", which is the only thing I know about that movie.Twelve / dozen12Twelve is a highly composite number - it has more divisors than any smaller number (for more on highly composite numbers see 36). Because of that, twelve has a special numerological significance, as sets of twelve are notably common in the human world. Some examples are:- 12 months in a year- 12 zodiac signs- clocks in America and sometimes in Europe are 12 hours- 12 inches in a foot- 12 eggs in a typical package- 12 days of Christmas- the 12 Olympic Greek gods- 12 function buttons on a keyboard- 12 notes in a scale in Western musicIn fact, sets of twelve are so common that such a set gets a special term called a dozen (for example, a box of a dozen donuts) - the name comes from French "douzaine" meaning twelve of something.Twelve has the property of being the smallest abundant number. An abundant number is a number whose factors add up to a number larger than the original. Twelve is abundant since its factors (1, 2, 3, 4, and 6) add to 16, which is larger than 12. Additionally, a perfect number is a number whose factors add up to exactly the original number (they're very rare, for more about them see 496) and a deficient number is a number whose factors add up tot a number smaller than the original. An interesting property of abundant numbers is that if a number is perfect of abundant, all of its multiples will be abundant. Therefore abundant numbers are common, as around 25% of numbers are abundant.Twelve is also the number of pentominoes (see also 35), and the fourth "weak factorial" (see that entry for more), i.e. the smallest number divisible by 1 through 4.The main prefix for twelve is dodeca-, which is Greek. It is found in words like dodecagon and dodecahedron. Its Latin prefix, duodeci-, is found in a few words like duodecimal, the usage of twelve as a numeral base instead of ten.(God, imagine how awesome it would be if we used a duodecimal system instead of a decimal system. A base divisible by 2, 3, 4, AND 6!)Thirteen / baker’s dozen13Thirteen has a number of mathematical properties: it is the sixth prime, a Fibonacci number (next is 21) and the fourth busy beaver number, the largest fully known value of the sequence. The first three busy beaver numbers are 1, 4, and 6 - for more on them see 107 and 4098. The prefixes for thirteen, fourteen, fifteen, etc. are trideca-, tetradeca-, pentadeca-, etc.13 is the number of Archimedean solids - those are 3-D figures that are fully symmetric like the five Platonic solids, but have two or more different types of regular-polygon faces. Those 13 solids are:- the truncated tetrahedron, the simplest which is just a tetrahedron with its corners cut off- the cuboctahedron, as its name suggests it combines properties of the cube and the octahedron with 8 triangular and 6 square faces- the truncated cube, a cube with the corners cut off which has 8 triangular and 6 octagonal faces- the truncated octahedron, an octahedron with the corners cut off which has 6 square and 8 hexagonal faces- the rhombicuboctahedron, this is my favorite one, which i like to think of as the octagon's 3-dimensional equivalent- the truncated cuboctahedron, a cool shape which looks kind of like the rhombicuboctahedron's more complex big brother- the snub cube, this one looks like a smoothed-out cube and has 6 square and 32 triangular faces- the icosidodcecahedron, this one combines the icosahedron and the dodecahedron with 20 triangular and 12 pentagonal faces- the truncated dodecahedron, which has 12 decagonal (10-sided) faces and 20 small triangular faces that look "chipped-off"- the truncated icosahedron, this one's shape is familar because it looks very much like a soccer ball- the rhombicosidodecahedron, this is has 62 faces and is somewhat reminiscent of the rhombicuboctahedron- the truncated icosidodecahedron, this one is to the rhombicosidodecahedron as the truncated cuboctahedron is to the rhombicuboctahedron- the snub dodecahedron, this is the most complex and the smoothest-looking one with 92 faces13 is also the smallest emirp - an emirp is a prime number whose digits, when reversed, produce a different prime, in this case 31. See also 17.Thirteen is perhaps most notable for having a connotation of bad luck in Western culture. This means that often, house numbers or elevator floors of 13 are omitted, and Friday the 13th is considered a very unlucky day. In fact, to keep an even-odd-even-odd pattern in numbering, sometimes even 14 is omitted in such numberings - for example, I have seen airplane seats that are numbered 1, 2, 3, 4, ... 10, 11, 12, 15, 16, 17, etc.13 of something is sometimes known as a baker's dozen, similar to the term "dozen" to twelve, but the term is a lot less widespread.It's also the current best lower bound to the problem where Graham's number arose - see my page on Graham's number for details.Fourteen / Poulter’s dozen1414 is an example of a semiprime, a number that is the product of two prime numbers, in this case the product of the primes 2 and 7. Semiprimes have no more than two non-trivial divisors (i.e. divisors that are not one and itself), and if the semiprime is a square then it has only one non-trivial divisor.Fourteen is the first even number that doesn’t really have a feel of completeness due to being the double of the somewhat irregular seven - it can't be divided by very many numbers, as its only factors are 1, 2, 7, and 14.14 is the smallest Keith number (see 197).I have heard of 14 of something being called a "poulter's dozen", though that usage is quite rare.Bignum Bakeoff was a competition held in 2001 where the goal was to make a C program with no more than 512 characters to produce as big of a number as you can. 20 entries were submitted, and 14 of them produced a value that was not 1. Of the other six, three of them (carnahan.c, pete.c, pete-2.c) didn't terminate, and three of them (dovey.c, edelson.c, f.c) only produced 1. The fourteen programs that did produce a large number were, in order of output (smallest to largest):pete-3.c, pete-9.c, pete-8.c, harper.c, ioannis.c, chan-2.c, chan-3.c, pete-4.c, chan.c, pete-5.c, pete-6.c, pete-7.c, marxen.c, loader.cClick on any one of the links above to see an entry on the list describing each of the programs sumbitted.Fifteen15Fifteen is an example of a triangular number, a number that is equal to the sum of the first x integers - in 15's case, 15 is 1+2+3+4+5. The nth triangular number can be calculated with the well-known formula, n*(n+1)/2. Triangular numbers are named that because a triangular number of things can be arranged as a triangle. For example, with 15 we have: o o o o o oo o o oo o o o oThe first 20 triangular numbers are:1 3 6 10 15 21 28 36 45 5566 78 91 105 120 136 153 171 190 21015 is the magic constant of the smallest and best-known magic square, a square of distinct numbers (none is used more than once) whose rows, columns, and diagonals add up to the same number. The smallest is the only (not counting rotations and reflections as distinct) 3x3 magic square, which is:4 9 23 5 78 1 615 is halfway between the numbers 10 and 20, and therefore one of the more common numbers from 11 to 19. Like 14, 15 is also an example of a semiprime. 14 and 15 are the first pair of semiprimes that are neighboring numbers - I call such numbers twin semiprimes. See also 33.Sixteen16Sixteen is an example of a perfect power, a number that can be expressed exactly with exponents and integers. Of perfect powers, numbers that can be expressed as x^2 are called square numbers/perfect squares, numbers expressible as x^3 are called cubic numbers, numbers expressible as x^4 can be called tesseract or quartic numbers, continue with penteract, hexeract, etc, with the Greek number prefixes.16 is 2^4 (second tesseract) and 4^2 (fourth square). Actually, it's a particularly interesting perfect power. It's the first number expressible as a perfect power in more than one way (see also 64, 81, and 4096). Better yet, it has the special property of being the only non-trivial solution of x^y = y^x where x and y are both integers. See also square numbers at 25.Sixteen is part of an interesting sequence of numbers of the form x^x^x. It is the second member of the sequence, preceded by the trivial 1 and succeeded by an astronomical number equal to about 7.6 trillion. For more on that sequence see my entry on 10^10^10.16 is another more common number because it’s a power of 2. Its prefix, hexadeca-, is fairly common, such as in the term "hexadecimal", the usage of 16 as a numeral base - that connects to computers' connection with powers of 2 (see my entry on 2 for more on that).In the googo- naming system, 16 can be named googoij. See also googoi (2) and googoiji (216).Seventeen17Seventeen is both one of my childhood favorite and current favorite numbers, and one with so long of an entry that it has its own page (READ MORE)Eighteen1818 is this list's example of a composite number, which you probably already know, is a number that can be evenly divided by numbers other than 1 and itself. In 18's case, 18 can be divided by 2, 3, 6, and 9. Its prime factorization is 2*32. Why did I choose to give an "example entry" for composite numbers? Because many of my list's entries are mainly examples for certain classes of numbers (Kaprekar numbers for 45, highly composite for 36, primorial primes for 29, sums of primes for 58), and thus I've decided to add example entries for things like composite and prime numbers for completeness's sake - Robert Munafo does this to a greater extent on his list.Because 18 is quite highly divisible for its size, it's a fairly commonly used number for grouping things. It's the second smallest abundant number, followed by 20, 24, 30, 36, 40, 42, 48....18 has the property of being the only positive integer that is exactly twice the sum of its digits. This is quite easy to see if you consider that if a number has three or more digits the sum of the digits will always be necessarily much less than half of the number, and then you can check each number below 100 individually to see this.18 is the most common adulthood age in the world, as in most countries the age you are considered an adult and the voting age is 18. Other common adulthood ages are 16 and 21.Nineteen19Nineteen is an example of a prime number, a number that can't be divided evenly by any numbers other than 1 and itself. Primes are significant in mathematics for zillions of reasons as I discuss on this list, and the first few are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37...Nineteen is the eighth prime number. It's also twin primes (primes two numbers apart) with 17, cousin primes (primes four numbers apart) with 23, and sexy primes (primes six numbers apart) with 13. This makes it an example of a number in all three well-known prime pairs. Such numbers are uncommon, and another example is 11. See also 23, 53, and 211.19 and 17 are an interesting prime pair; they're twin primes that are both exponents of Mersenne primes (131,071 and 524,287 respectively). 17 and 19 appear to be the largest twin prime pair to have this property.Twenty20o o o o oo o o o oo o o o oo o o o o^ For visualization, that above is twenty o's.Twenty is an example of a tetrahedral number - a tetrahedral number is a number that is equal to the sum of the first n triangular numbers (see 15). The first few are 1, 4, 10, 20, 35, 56, 84, 120, 165, 220... and they're called tetrahedral numebrs because a tetrahedral number of spherical objects (e.g. oranges) can be arranged in a tetrahedron (aka a triangular pyramid):layer 1 layer 2 layer 3 layer 4 o o o oo o o o o oo o o o o o o o o oLike with triangular numbers, tetrahedral numbers have a special formula: the nth tetrahedral number is equal to n*(n+1)*(n+2)/6.Twenty has sometimes been used as a base of number naming. This may be because twenty is the total count of all fingers and toes on a human being. For example, the Mayans used twenty as their base, with five as a sub-base, and 20 is also used in traditional Welsh and in some African languages, and in Danish numerals from 50-99 and in French numerals from 60-99.Twenty of something is sometimes called a score (see the entry for 87 for an example of this usage), though this usage is somewhat dated nowadays.The most common prefix for 20 is icosa- (from Greek), like in icosahedron. The Latin prefix for 20, viginti-, is sometimes used in terms like vigintillion, a favorite large number of mine.20 is the number of entries that were submitted in Bignum Bakeoff, a large number competition in 2001 where the goal was to make a program in C to generate the largest number you can. For more on it see 14.Twenty-one2121 is notable in the English numeral system for being the first number name constructed from existing names. This is somewhat important in googology because the term "googolism", albeit a broad term, more often than not refers to a specific unique name for a number, and twenty-one is the smallest nonnegative integer that does not get a unique name in English - therefore in some sense twenty-one is the smallest nonnegative integer name in English that is not a googolism. Sbiis Saibian has further discussed the meaning of the term "googolism" on some of his Googology Wiki blog posts. See also grangolplex/googolcentiplex.21 is also the 7th Fibonacci number, preceded by 13 and followed by 34 - for more on them see 89. It is also the value of S(3) in the frantic frog function, a variant of the busy beaver function in googology. The S function is noted S(n) (for more on it see 107), while the busy beaver function is noted Σ(n) or BB(n).Twenty-two / Dumevalka2222 is the smallest Lychrel number in base 2, i.e. the smallest number in binary that will never become a palindrome if you take the number, add its reverse, and repeat the process - see 196 for details. In bases that are powers of two it's easy to see that some numbers are Lychrel numbers. In base 10 that has not been proven, though there are some candidate base-10 Lychrel numbers, the smallest and best known of which is 196.Twenty-two is the exact value of a very small googolism coined by Googology Wiki user SpongeTechX using a notation he invented called copy notation. The number is called dumevalka, and it's defined as 2[2,2]. This solves solves to 2[[2]] (a[b,c] = a[^bc]b, where [^b is [[..[[ with b [s) = 2[2] (a[^bc]b = a[^b-1a[^b-1a[^b-1.......[^b-1a]^b-1.....]^b-1 nested c times, so this is 2[2] with two nestings) = 22 (a[b] = aaa.....aaa written as a number, with b copies of a, so this is two copies of two). It's an example of degenerate cases produced by two in googological functions, and in fact it mirrors the degenerate case of 4 in up-arrows, Conway chain arrows, and Bowers' arrays.22 is also (as of August 2014) the smallest value for which the Busy Beaver function is known to surpass Graham's number - that is, BB(22) is known to be greater than G, but BB(21) and below may or may not be. Previously the smallest such value was 24; before that it was 25, and before that it was 64.Twenty-three2323 is the ninth prime number. It's notable for being the first one not to be part of a pair of twin primes (primes two numbers apart) or closer, since 21 and 25 are both composite. Such prime numbers are called isolated primes, but once you get to large enough numbers, the majority of primes will be isolated primes.Also, 23 is an example of a factorial prime, a prime one more or less than a factorial number, similar to primorial primes (see 29). The sequence of factorial primes begins 2, 3, 5, 7, 23, 719, 5039, 39,916,801....23 is the smallest prime number with the property that you can remove any digit and it's still prime - for more on that kind of thing see my entry for 137.Additionally 23 is the number of chromosome pairs in a normal human cell (see also 46).23 is also a cult number noted largely for Illuminati associations and conspiracy theories, and a number I seem to randomly produce pretty often. For example, 23 is randomly chosen in the entry for 1, and a draft version of this list had -23 on the list but specifically excluded 23 (sorry people who like the number 23). Therefore 23 is my second favorite cult number behind 17.Gaz23 ⅔See 10^74.Twenty-four2424 is an example of a factorial number, a number that is the product of the first n positive integers. It's the fourth factorial, equal to 4*3*2*1. The nth factorial is denoted n!, and the sequence of factorials grows pretty quickly, starting with 1, 2, 6, 24, 120, 720, 5040, 40,320 ... Factorials have many applications in mathematics, in combinatorics and elsewhere. The most well-known is that n! denotes the number of ways you can arrange n objects in order - for example here are the 24 (4!) ways to arrange 4 objects:ABCD ABDC ACBD ACDB ADBC ADCBBACD BADC BCAD BCDA BDAC BDCACABD CADB CBAD CBDA CDAB CDBADABC DACB DBAC DBCA DCAB DCBAThe factorial is a classic example of what the non-googologists see as a fast-growing function, and therefore it's part of the layman's toolbox for generating large numbers. Some googolisms are defined with factorials like the googolbang, and Lawrence Hollom has developed a powerful (yes, powerful even by googology standards) notation to extend the factorial function into the realm of googology.24 is also a highly composite number since it has more divisors than any smaller number. Its non-trivial divisors are 2, 3, 4, 6, 8, and 12.There are quite a lot of things that are grouped into 24, similarly to twelve. For example, there are 24 hours in a day, food and bottles are often packaged in groups of 24, sports teams often come in groups of 24, and purity of gold is measured on a scale of 0 to 24 carats.24 is the name of a popular game where you are given four single digit numbers and you need to use three steps with addition, multiplication, subtraction, and division to turn them into 24. The choice of 24 was probably motivated by the fact that 24 has a lot of divisors for its size.Personal: For a time, when I didn't have a solid decision on my "favorite number", 24 was my favorite number (probably because of the game) - I switched back and forth between several numbers before settling on a childhood favorite as my "favorite number", 7.Twenty-five2525 is the fifth square number (5^2), and this list's example of a square number. Square numbers are numbers that can be expressed as a number multiplied by itself, or as x^2 (pronounced x squared), and are named like that because square numbers of objects can be arranged in a perfect square - for example here's a square of 25 o's:o o o o oo o o o oo o o o oo o o o oo o o o oThe first 20 square numbers are:1 4 9 16 25 36 49 64 81 100121 144 169 196 225 256 289 324 361 400See also cubic numbers at 125 and perfect powers in general at 16.25 shows up quite often in life, being a quarter of 100 (see also 50). For example, many currencies have a 25-cent coin, and 25% is one of the most used percentages (see 1/4).25 is the smallest square that is a sum of two squares: 25 = 5² = 3²+4². However, this property isn't too special - it's more interesting that 25 is a square that is a sum of two consecutive squares because that property is rare. The first few squares with that property are 25, 841, 28,561, 970,225....25 is also the smallest (and only two-digit) Friedman number. A Friedman number is a number whose digits can be rearranged with mathematical operators to produce the original number. In 25's case, 25 = 5^2.Twenty-six2626 is the only number that falls directly between a perfect square (25) and a perfect cube (27). It also might be the one and only number that falls directly between a pair of perfect powers, though this hasn't been proven - proving that kind of thing is quite difficult. For example, not until 2002 was it proven that 8 and 9 are the only pair of perfect powers that neighbor. It's been conjectured that for any integer n there's only a finite number of pairs of perfect powers that differ by n, though as of yet that hasn't been proven.26 is also the number of letters in the Latin alphabet. They are abcdefghijklmnopqrstuvwxyz, making 26 somewhat culturally significant, such as in combinatorics with things like codebreaking. Powers of 26 also appear sometimes in everyday combinations, e.g. 26³ = 17,576 possible three-letter acronyms.Because there are 26 letters in the alphabet, 26 had some significance to me as a kid, since when I was 1 to about 6-7 years old the alphabet was pretty much my favorite thing in the world.(pointless fact: 26 is called "lel" in Sbiis Saibian's experimental unique name system for numbers, which pretty much explains why such systems don't work. This connects to me because for a while I tended to use "lel" in place of "lol". By the way, 28 is called "lol" in that system.)Twenty-seven / Boogathree / Fzthree2727 is equal to 3^3, the third cube. It can be defined with tetration as 3^^2 but is much too small to even be a small tetrational number, not even a good exponential number. Related to being the third cube, it's the smallest odd composite number that is not a semiprime.27 has an interesting property relating to cubes - the digits of 27^3 (19,683) add up to 27 (1+9+6+8+3 = 27). 27 is the largest number with this property, and 8 is the smallest (not counting the trivial cases of 0 and 1). - 17, 18, and 26 also have this property. It's especially interesting that 8 and 27, the smallest and largest number with this property, are themselves cubes.27 used to have a notable property in Bowers' array notation that is now held by a googolism called tritri - {3,3,3} in Bowers' array notation once was 3^3, but it's now equal to 3^^^3 since Chris Bird suggested to make {a,b} solve to a^b instead of a+b. See ultatri for an example of 27's occurrence in Bowers' arrays. 27 is still is a very prominent number when working with threes in googology (example: 3^^^3 is a power of 3^27 threes), especially often in chained arrow notation (see part 2 of this list for examples). This is because the quirks of chained arrow notation give rise to exponents in the middle of each other - for example, 3->3->2->2 solves to {3,3,27} in Bowers' arrays. See my article on Conway chain arrows for further information.With the fz- prefix which takes a number to the power of itself, 27 can be named fzthree. Sbiis Saibian coined a prefix called booga-n, meaning n^^n-2n, or n n-ated to n in terms of hyper-operators. With that function, 27 can be represented as booga-three. For more on the booga- prefix read here. See also booga-four.27 is also another cult number, and it's Robert Munafo's favorite as he discusses on his number list - he says that he "notices it a lot more than he 'should'" and puts some 27-based numbers on his list just for the sake of 27-based numbers. These three links are a few 27-based cult number websites, and this page (which gives a far far better covering of 27 than my entry does) might as well count as one. An example of the cult of 27 is the 27 club in music - the 27 Club consists of the many musicians (like Jimi Hendrix, Jim Morrison, Kurt Cobain, Amy Winehouse, etc, etc) who died at the age of 27, giving rise to some speculation. 27's cult status actually dates all the way back to Roman times where it was important in certain places.Personal: 27 is part of the product 41*27Twenty-eight2828 is the second perfect number. It's a perfect number because it's equal to the sum of its factors, which are 1, 2, 4, 7, and 14. The previous perfect number is 6 and the next is 496 (see that entry for more on perfect numbers).28 is also used as an approximation for the number of days in a lunar month, which matches nicely with the designation of 7 days in a week.Twenty-nine2929 is the tenth prime number. It's twin primes with the Mersenne prime 31, and it's a primorial prime as well. A primorial prime is a number one more or less than a primorial (see 30 for more).29 is the number of notches on the Lebombo Bone, a 35,000-year-old bone in Africa that was the first known documentation of numbers. It was clearly used for counting something from the notches in the bone. For more on the bone see my large number timeline.It's also the smallest number x such that Graham's number cannot be written as a power tower of x's (not)Thirty30o o o o o oo o o o o oo o o o o oo o o o o oo o o o o oThirty is the product of the first three prime numbers (2*3*5). Products of the first n prime numbers are known as primorials, and n primorial (the product of all primes less than or equal to n) is sometimes denoted n#.30 is a common number for dividing things evenly, since it can be divided evenly into twos, threes, fives, sixes, and tens, and it forms two primorial primes (29 and 31). However it's not a highly composite number.Thirty is also the largest number with the property that all smaller numbers that are relatively prime to it (i.e. have no factors in common with 30) are prime.Thirty-one31Thirty-one is a prime number with several interesting properties. It is perhaps most notable as the third Mersenne prime (M₅ = 2^5-1), since Mersenne primes are known to be the easiest way to find very large prime numbers. It's also a primorial prime, along with 29.31 is a member of the primeth recursion sequence, a sequence defined as PR(1) = 1 and PR(n+1) = the nth prime number. The first few values of the primeth recursion sequence are 1, 2, 3, 5, 11, 31, 127, and 709. The sequence is the nth term sequence for the prime numbers - see my page on the Ackermann function (which is closely related to such sequences) for details. Interestingly, two consecutive members of that sequence (31 and 127) are both Mersenne primes.31 is the starting member of a sequence of similar-looking prime numbers that starts with 31, 331, and 3331, continues like this, and ends with 33,333,331. A similar sequence notable for being pretty long is discussed in the entry for 43.31 also one of only two numbers that is a more-than-2-digit repunit prime in more than one base: it is 11111 in binary and 111 in base 5. The only other number with this property is 8191.Thirty-two / Binary-eyelash mite32This number, due to being a power of two (second penteract), has sort of a digital feel to it. This is because computers run on binary digits, and therefore basing things on powers of two is often most convenient. A direct example of 32 in computing is the system32 folder and 32-bit.32 is the third member of the sequence 1¹+2²+3³ ... +nⁿ, expressible as 11+22+33. 32 seems to be the largest perfect power in the sequence, but honestly that's just a guess.32 is also notable in science as the freezing point of water in degrees Fahrenheit. For more temperature related numbers, see -40, 37, 98.6, 212, and 273.32 is the exact value of a very small googolism by Sbiis Saibian - it's called binary-eyelash mite. Since an eyelash mite is defined as 2*10^4, a binary-eyelash mite can be thought of as being 2*2^4, which evaluates to 32. See binary-small fry, another small googolism by Saibian, for more on that.Thirty-three3333 is the smallest number in the first semiprime triplet (as I call it): a semiprime triplet is a set of three neighboring integers that are all semiprimes: 33 = 3*11, 34 = 2*17, 35 = 5*7. The next semiprime triplet is (85, 86, 87), followed by (93, 94, 95) and (121, 122, 123). Semiprime quadruplets are impossible because one of the four consecutive numbers must be divisible by 4, which means that it's necessarily a number that is not a semiprime. For similar groups but with prime numbers, see 47, 211, and 251.As a consequence of Fermat's Little Theorem, if p is a prime other than 2 and 5, (10^(p-1)-1)/p is a whole number. 33 is the smallest number that can be generated with that formula: here, p = 3, so plugging 3 in gives us (10^(3-1)-1)/3 = (10^2-1)/3 = (100-1)/3 = 99/3 = 33. This formula sometimes forms cyclic numbers (33 is not cyclic) - for more on that, see 142,857.Thirty-four3434 is the eighth Fibonacci number. It's the middle number in the first semiprime triplet (see 33), and as such it's the smallest number with the property that it and its neighbors have the same number of divisors. It is also known as part of an Internet meme, Rule 34, which states that "if it is on the Internet, there is porn of it, no exceptions". Similarly, Rule 34 of googology (jokingly stated by Sbiis Saibian) states that if it is a googolism, there is a salad number for it. A few salad numbers are on later parts of this list.Thirty-five3535 is the the largest number in the first semiprime triplet (see 33). It's also an example of a 5-rough number. N-rough numbers are the antithesis of n-smooth numbers, because while a n-smooth number only has prime factors n or smaller (e.g. 48 and 81 are 3-smooth numbers), a n-rough number only has prime factors n or larger. 3-rough numbers are just a synonym for odd numbers, and the first few 5-rough numbers are 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43... - notice how a lot of them are prime numbers.Also, 35 is the number of hexominoes (polyominoes made fron six squares, picture below).Thirty-six3636 is 6^2, the sixth square number. It's also the 8th triangular number and the smallest number (aside from the trivial case of 1) to be both a square number and a triangular number. Such numbers are called square triangular numbers, and they are quite rare: the first few are 1, 36, 1225, 41,616, 1,413,721...36 is also an example of a number which is highly divisible, as it can be divided by two twice and by three twice - its nontrivial factors are 2, 3, 4, 6, 9, 12, 18. It's a record-setter in the number of factors, being the first one to reach seven non-trivial factors. These record-setting numbers are called highly composite numbers. The first few such numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 360, 720, 840, 1260, 1680, 2520, 5040... and they tend to be very popular numbers for dividing things in (e.g. 24 hours in a day, 60 seconds in a minute and minutes in an hour, 360 degrees in a circle) - see also 12, which a famously large number of things are grouped in.36 is the largest highly composite number to be a perfect power - the only other one (other than the trivial case of 1) is 4.Thirty-seven3737 is the twelfth prime number. It's also equal to hydra(3), the last trivially sized value of the Kirby-Paris hydra function. The hydra function is defined using a single-player game which has a few simple rules:- Start with a finite sequence of matched parentheses such as (()(()(())((())))).- Pick an empty pair () and a natural number n. 1. Delete it. 2. If its parent is not the outermost pair, take its parent and append n copies of it.The parent of a parentheses pair is the innermost pair of parentheses that encloses the pair and fully encloses all parentheses within the pair, and all that is within that pair. For example the parent of the bold pair in (()(()(())((())))) is the the red parentheses in (()(()(())((())))).Then hydra(n) is defined as the number of steps it takes to reduce a hydra (((...((()))...))) with n pairs of (), always picking the rightmost pair of () and using 1 as n on the first move, 2 on the second move, etc. Hydra(0) = 0, Hydra(1) = 1, Hydra(2) = 3, Hydra(3) = 37, and Hydra(4) and beyond are very very large.Unlike the busy beaver function, the hydra function is computable, with a growth rate of epsilon-zero in the fast-growing hierarchy. See also 51.1/37 is 0.027027027.... and 1/27 = 0.037037037.... - this nice relationship is because 27*37 = 999.Thirty-seven was another entry in my "Very Important Numbers" list. This one was because 37 multiplied by 3, 6, 9, 12, etc. produced 111, 222, 333, 444, etc. Mathematicians would find this property trivial, but I found it cool at the time that such a random (37 is psychologically random) number could produce repdigits (and that's also related to the correspondence between 1/27 and 1/37). 37 remains one of my favorite two-digit numbers, though I don't seem to pay attention to it as much as I do for 17, 43, and 79.Because 37 is a factor of 111, it has an interesting divisibility test: 1. If the number is longer than 3 digits, divide it in groups of 3 digits starting from the right, e.g. 47,356,892 -> 47, 356, 892 2. Add the groups together, e.g. 47+356+892 = 1295 3. If the result has more than 3 digits, repeat the process - doing that gives you 296When the number no longer has more than 3 digits: 1. If it's 1-2 digits, it's divisible by 37 if it's 37 or 74 2. If it's a 3-digit repdigit, then it's divisible by 37.3. Otherwise, subtract the number from the nearest multiple of 111, and if the result is 37 or -37 then the number is divisible by 37. Here, we find that 333 is the closest multiple of 111 to 296, and 333-296 = 37. This means that 47,356,892 is divisible by 37.In science, 37 is the normal body temperature in degrees Celsius. For other temperature related numbers, see -40, 32, 98.6, 212, and 273.I consider 37 to be the quintessential psychologically random number because 37 is said to be the number people are most likely to choose when asked to name a number from 1 to 100. It's one of the survivors when we filter out the non-random numbers with this process:First we can remove all the single-digit numbers, leaving 91 numbers left. Then, we can delete the multiples of 10, leaving 81 left. After that, we can get rid of the even numbers, removing 36 numbers and leaving 45. While we're at it we can remove the repdigits as well, leaving us with 40 numbers, Even the numbers that end in 5 can be removed, leaving left 32 numbers (13, 17, 19, 21, 23, 27, 29, 31, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 79, 81, 83, 87, 89, 91, 93, 97).To narrow down further, we'll need to make further observations: the remaining numbers all end in 1, 3, 7, and 9. Of these, the most random ending digit seems to be 7, leaving us with 17, 27, 37, 47, 57, 67, 87, and 97. Of those, we'll now want to focus on size. We use and think about the smaller ones in our daily life more often than the larger ones. The larger numbers are less likely for us to think of, and the smaller ones are so common that they don't seem random. 37 is just the right combination of "large enough to be random" and "small enough to be random". And that, is how you get 37 as the "most random" number (though some give the honor to 47).As it turns out, 37 is a cult number as well, like on this website. The property I noted in my Very Important Numbers list happens to be one of the many many properties listed on the website.Thirty-eight3838 is an example of a psychologically random even number, for reasons similar to 37. It's also part of a twin semiprime pair with 39. 38 is also the largest even number that can't be written as a sum of odd composite numbers.When writing each number that can be expressed in Roman numerals (i.e. numbers 1 to 3999) and alphabetizing them, the last Roman numeral would be 38 (XXXVIII).Thirty-nine3939 is infamous as part of a claim that 39 is uninteresting: it was claimed to be the "smallest uninteresting positive integer, which makes it especially interesting" in the first edition of David G. Well's book The Penguin Dictionary of Curious and Interesting Numbers (a list of numbers with notable properties much like Robert Munafo's). In response, some people gave connections between 39 and the famed number 666, but that isn't all that special since so many other numbers have been connected to 666 (like 1998). But Wells wasn't aware that 39 did indeed have an interesting property.What is that property? 39 is the sum of a sequence of consecutive primes (3+5+7+11+13) and the product of the first and last numbers in the sequence (3*13). Very few numbers have this property - the only other numbers known to have that property are 10, 155, 371, and 4,545,393,575,304,421. That property was noted as early as 1999, in an old version of Robert Munafo's number list which was a lot shorter than his modern list.In the second, revised and expanded edition of his book, Wells added that property to 39, and his first uninteresting number is instead 51. As a side note, the largest number in his book is Graham's number. (What were you expecting?)Forty40o o o o o o o o o oo o o o o o o o o oo o o o o o o o o oo o o o o o o o o oThis number is close to the limit of our secondary number sense. It’s getting a bit tough to accurately visualize, say, forty people in a bus. This number is spelled differently from what we might expect, “fourty”. This number’s prefix is tetraconta-, but by now you won’t see these prefixes outside of lists. Take a guess what the prefixes for fifty, sixty, seventy, eighty, and ninety are.Forty appears many times in religion, particularly with periods of forty days. For example, Lent is noted as the forty says before Easter. It also appears in the phrase "forty acres in a mule" which refers to what freed slaves hoped for after the Civil War, a promise that was unfulfilled.Forty is also the only number whose English name has the letters in alphabetical order.Forty-one4141 part of an interesting formula discovered by Leonhard Euler (x^2+x+41) that will always output a prime number when x is between -40 and 39, inclusive, and for some other values as well. Another way to think of the formula is:41 is primeAdd 2, 43 is primeAdd 4, 47 is primeAdd 6, 53 is primeAdd 8, 61 is primeAdd 10, 71 is primeAdd 12, 83 is primeetc, you can continue the pattern 39 times.This is interesting because the formula, unlike others which are a bit more wide ranged, is easy to remember. In fact, it's the most widely ranged prime-generating polynomial that is easy to remember. After this formula was discovered, people wondered if there were polynomials that would always generate primes for any integer input - this was proven impossible.41 is also the next prime after 19 to be part of a twin, cousin, and sexy prime pair - not a lot of prime numbers have this property.Personal: 41 is part of the product 41*27Phiplex10^((1+√5)/2)~ 41.4986...Since x-plex can be thought of as 10^x because a googolplex is 10 to the power of a googol, we can call 10^phi (phi is the golden ratio) phiplex - it's a relatively small number (less than 100), but can be considered a googolism nonetheless.Forty-two42Forty-two is famous for being calculated as the Answer to the Ultimate Question of Life, the Universe, and Everything in The Hitchhiker’s Guide to the Galaxy—a book I must sadly admit I haven't read—which many people interpreted as the meaning of life or something similar. Therefore, 42 is a very culturally significant number. It's an example of a number whose reputation has been swamped into one strong association due to pop culture. As a result, 42 is a famous cult number - as with numbers like 666, many people have tried to use real-world connections to 42 to deduce why 42 is the "meaning of life".Despite all those theories about 42, Douglas Adams, author of the book that made 42 famous, said that all those theories are nonsense, and that he only chose 42 because it is an "ordinary, smallish number" - he thought of "42" as an ordinary smallish number to use and thought that'll do. But I think a particular reason he chose 42 is because 42 is a random number, but not too random like the psychologically random numbers 17 or 23 or 47, which already were large cult numbers - psychologically random numbers and cult numbers are strongly related. If the "meaning of life" was something like 17, then the cult of whatever number it would've been in the book would likely have a very different story.Before Adams, Lewis Carroll also referenced 42 several times in his writing, but he did not manage to get 42 to be a cult phenomenon like Adams did.For another number that appears in The Hitchhiker's Guide to the Galaxy, see the Hitchhiker's Number.Forty-three4343 is a prime number that appears in an interesting sequence called Sylvester's sequence, a sequence that starts with 2 (or alternately 1, doesn't make a difference to the sequence), and each member of the sequence is the product of all the previous terms plus 1. It begins:1, 2, 3, 7, 43, 1807, 3,263,443, 10,650,056,950,807...so the sequence grows very quickly, faster than even the factorial. In fact, it achieves hyper-exponential growth rates, which basically means that the number of digits grows exponentially - in this function's case, the number of digits in any term of the sequence is about twice the number of digits in the previous term. Another function that achieves hyper-exponential growth is the fuga- function.It is fairly easy to see that all the terms have no prime factors in common, and the sequence has been used in one of the many many proofs that there are infinitely many prime numbers. Another such proof involves showing that all the Fermat numbers have no prime factors in common.Another interesting thing about the sequence is that the reciprocals of its members (not counting 1) add up to one: 1/2+1/3+1/7+1/43+1/1807.... = 1. It's the fastest growing sequence whose reciprocals converge to a rational number.43 can also start a sequence of primes by adding digits either from the right (43, 439, 4391, 43,913, 439,133, and 4,391,339) or from the left (43, 443, 8443, etc, up to 6,933,427,653,918,443), and is a record setter for length of said sequence when growing from the left (for two-digit numbers that is).43 was also one of my childhood favorite numbers, and one of my most loved two-digit numbers as a kid (see also 79). This one's story is kind of weird: I believe it came from a VHS (see also 1107) to teach kids about math I watched several times in first grade. In one scene, the narrator listed the names of some prime numbers, and as he narrated each one, the number appeared in a different part of the screen, and it ended up being filled with about twelve primes. 43, for some reason, struck me as a funny number that I should pay attention to, probably because it was the last one they listed and/or because it's psychologically random, which always makes a number appealing. From then on I have often compulsively checked lists just to see what the 43rd entry is (ok I admit it, I still sometimes do that), among other things with the number 43.xkcdBooga-pi with Nayuta Ito's system~43.825904Remember booga-e, equal to about 10.30495? Well, with the same generalization of up-arrows you can make values like booga-pi, equal to pi{pi-2}pi, with {pi-2} indicating pi-2 (or about 1.14) up-arrows, which can be imagined as a hyper-operator just beyond exponents.Calculating booga-pi with that system is a lot harder than calculating booga-e, so I ran a program to calculate the value and got 43.825904 as the result.Forty-four4444 is an example of a subfactorial number, and the fifth one. The subfactorial (sometimes denoted !n) is a function similar to the factorial but not as quickly growing - it represents the number of ways n objects can be ordered such that none of them are in their original place. The sequence of subfactorials begins 1, 2, 3, 9, 44, 265, 1854....For example, four subfactorial is equal to nine: the nine arrangements for four objects (A, B, C, D) such that none are in their original place are:BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB, DCBAA way to imagine the subfactorial in real life is like so: Imagine that there are five people, each with one hat, who meet together to trade hats. Each person wants to end up with a different hat after the trading is over. Then, there are 44 ways this could happen, i.e. 44 ways such that in the end nobody has their original hat.Curiously, you can calculate n subfactorial easy with the surprisingly elegant formula [n!/e] where [n] is the nearest integer to n.For another weaker version of the factorial see 420.Forty-five4545 shows up sometimes when working with degree measures of angles in geometry, because a 45 degree angle is halfway between a right angle and a fully closed angle. That angle is equivalent to slopes halfway between horizontal and vertical, and an example of such a slope is the slope of the function y = x.More notably, 45 is what Robert Munafo calls "the quintessential Kaprekar number": Kaprekar numbers are a special type of number best explained with an example: 45^2 = 2025, 20+25 = 45.Kaprekar numbers are numbers that have this property for squares. 999 is another Kaprekar number: 999^2 = 998,001, 998+001 = 999.45 is also a Kaprekar number for cubes (45^3 = 91,125, 9+11+25 = 45) and fourth powers (45^4 = 4,100,625, 4+10+6+25 = 45). In fact it is the only known number that is a Kaprekar number for squares, cubes, AND fourth powers.The first few Kaprekar numbers are 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, and 9999 - notice how they tend to be repdigits.45 is also one of the numbers that appears in the ending loop when performing Kaprekar's routine (see 6174) on two-digit numbers - the loop is 45, 9, 81, 27, 63, back to 45, 9, 81, etc.Forty-six46The largest even number not expressible as a sum of two abundant numbers. Also the number of chromosomes in a typical human cell - 22 pairs of normal chromosomes and two sex chromosomes, with 23 pairs.Forty-seven47Forty-seven is part of an equally spaced group of 3 consecutive primes (47, 53, 59). That triplet is the smallest one after 3, 5, and 7; the next is 151, 157, 163; and a little after, such triplets become more common. See also 33, 211, and 251.47 is also a strictly non-palindromic number. A strictly non-palindromic number is a number n that is not a palindrome (reads the same forwards and backwards, like 14,641) in any numeral base b, where 2≤b≤n-2 (that is, in any base from binary to base n-2). That means that in any base from binary to base 45 and all bases in between, 47 won't be a palindrome. The first few strictly non-palindromic numbers are 1, 2, 3, 4, 6, 11, 19, 47, 53, 79, 103, 137, 139....47 is yet another cult number (for being psychologically random), one large enough to be an in-joke in some places. A few of its cult websites are this one and this one - one calls it the "quintessential random number", but I think 37 deserves that title more. The second website I listed seems to be for a university that has connections to 47 and lists other places where 47 is an in-joke.Forty-eight4848 is the number of known Mersenne primes, as of October 2014. It's also notable for having a lot of factors (2, 3, 4, 6, 8, 12, 16, 24), and it's another highly composite number.48 is an example of a 3-smooth number, a number that has no prime factors greater than three. The first 3-smooth numbers are 1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 24, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216...... In fact, it's a member of a sequence of 3-smooth numbers of the form 3*2^n that starts 3, 6, 12, 24, 48, 96, 192, 384, 768...... these numbers all have a pretty large amount of divisors, and of these, 6, 12, 24, and 48 are also highly composite numbers. Therefore they occur in fields like display resolutions, e.g. 1024x768 my own desktop resolution. However no 3-smooth numbers larger than 48 are highly composite numbers.See also log(3).Forty-nine49Forty-nine is the seventh square number. It's also the smallest number with the property that it and its neighbors are squareful (divisible by a square number other than 1): 48 = 12*2², 49 = 7², 50 = 2*5².49 is sometimes given a connotation of luck since it's the square of a number (7) well-known for being considered lucky. For example, in Chinese culture the number 7 symbolizes togetherness, and therefore 49 is sometimes associated with good luck as well and used in some Chinese rituals.In the second, revised and expanded edition of his book, Wells added that property to 39, and his first uninteresting number is instead 51. As a side note, the largest number in his book is Graham's number. (What were you expecting?)Fifty50o o o o o o o o o oo o o o o o o o o oo o o o o o o o o oo o o o o o o o o oo o o o o o o o o o = fifty o’sNow this number is about the ending of our secondary sense and the beginning of our tertiary sense. Tertiary number sense allows us to roughly visualize numbers and somewhat approximate them, but it’s significantly less accurate than our secondary sense.Fifty shows up a lot in real life in general, being half of one hundred (see also 25). For example, it's the number of states in the United States, and it's been that way since 1959. That value is unlikely to change soon, as it would be odd to deviate from that round number. Additionally there is the fifty-move rule in chess which states that if the game goes fifty moves (a move is a turn by white followed by a turn by black) without a pawn move or capture, the game ends with a draw. See 14,296.50 is also the smallest number that can be expressed as a sum of two squares in two different ways: 5²+5² = 1²+7² = 50. See also 17, 65, and 1729.Fifty-one51The book where 39 was claimed to be uninteresting, in its second edition, instead has 51 as the first uninteresting number. For more on that see 39.51 is notable in googology for being the value of Worm(2) - Worm(n) is a fast growing function in googology (growth rate epsilon-zero in the fast-growing hierarchy) invented by Lev D. Belkemishev. It is closely related to the Kirby-Paris hydra (see 37), but a little more complicated. The worm function is defined as how many steps it takes to reduce a chain of integers [n] down to an empty chain using a specific set of rules. Those instructions are made in such a way that it will keep on expanding unless conditions allow you to remove a value - therefore it can take very long to reduce a chain to an empty chain, but it will always take a finite number of steps. A chain [1] takes 3 steps to reduce to an empty chain, so Worm(1) is 1, and a chain [2] takes 51 steps to reduce to an empty chain. Not much is known about Worm(3) and higher values, except that they're very very huge.Fifty-two52Fifty-two is the number of playing cards in a traditional deck: four suits (diamonds, hearts, club, spade) multiplied by thirteen faces for each suit (ace, 2 through 10, jack, king, and queen). Therefore it has a bit of numerical significance, and shows up sometimes when working with combinatorics in card games (see 649,740).52 is also usually given as the number of weeks in a year - that's an estimate, since dividing a year into 7 day weeks gives 52.14 weeks, 52.28 if it's a leap year. Fifty-three5353 is notable for being the first prime not to be part of a twin or cousin prime pair. Its two closest primes are 47 and 59, both of which are sexy primes with 53. The first prime not to be part of a twin, cousin, or sexy prime pair is my favorte 3-digit number 211.53 is said to be a magic number relating to birthdays - the chance that no two of any fifty-three people in a room share their birthday is approximately 1 in 53.And 53 is also the number of bits used to represent the part before the exponent in the commonly-used double floating point format in computing - for more on that see the entry on the maximum value of that format, 1.7976*10^308.Fifty-four5454 is the number of stickers on a traditional 3x3x3 Rubik's Cube - therefore, it shows up sometimes in combinatorics with Rubik's cubes. This leads to some pretty big numbers, such as the number of ways to arrange the stickers of a Rubik's cube.Fifty-four is also the number of words in the English language used to name integers. They are:negative hundredzero ten thousand decillion vigintillionone eleven million undecillion googoltwo twelve twenty billion duodecillion centillionthree thirteen thirty trillion tredecillion googolplexfour fourteen forty quadrillion quattuordecillionfive fifteen fifty quintillion quindecillionsix sixteen sixty sextillion sexdecillionseven seventeen seventy septillion septendecillioneight eighteen eighty octillion octodecillionnine nineteen ninety nonillion novemdecillion(click on any number above to go to its entry, except for 80 which doesn't have an entry so it links to 81 instead)Up to 10^66-1, the naming is unambiguous, but after that it gets a little more uncertain - for a more on that, see the entry for a vigintillion.Fifty-five55Fifty-five is the ninth Fibonacci number. The next one is 89. It's also the tenth triangular number (sum of the integers 1 through ten), and it's the largest number to be both a Fibonacci number and a triangular number.Fifty-six56A Latin square is a x-by-x grid where each slot is filled with one of x numbers (or letters or colors or symbols or anything, doesn't matter) such that each row and has each number/symbol exactly once. For example,4 3 1 22 1 4 31 2 3 43 4 2 1is a Latin square.A reduced Latin square is a Latin square such that the first row and column have the numbers ordered. By rearranging the rows and columns, any Latin square can become reduced. For example, the square above can become a reduced Latin square if we arrange the rows and columns differently:1 2 3 42 1 4 33 4 1 24 3 2 156 is the number of reduced 5x5 Latin squares - after that term, the number of reduced x*x Latin squares grows quickly, with 9408 reduced 6x6 Latin squares and 16,942,080 reduced 7x7 Latin squares. The number of non-reduced x*x Latin squares grow even faster.Latin squares are notable because of the well-known game Sudoku: all rows and columns in a Sudoku grid must have the each number 1 to 9 exactly once (with the added 3x3 box restriction). There are 6.67 sextillion possible Sudoku grids, and 5.524 octillion possible 9x9 Latin squares (Sudoku grids without the 3x3 box restriction). Also these squares are sometimes used in artistic designs, with different colors instead of numbers.56 is also the largest number you can raise to its own power on most scientific calculators which overflow when reaching a googol. See also 69 and 449.Fifty-seven5757 is the first integer that, when raised to its own power, makes a number larger than a googol.The number 57, for being a random-sounding number, has sometimes been jokingly referred to as a prime. For example, it's been called the Grothendieck prime because in a certain story, mathematician Adam Grothendieck gave 57 as an example of a prime number. It isn't acually a prime, as its factorization is 3*19. It is a semiprime though.57 is also a number associated with Heinz ketchup (and sort of a cult number among its fans): H. Heinz, the founder, promoted his ketchup by lying that there were 57 varieties, and the number 57 is still printed on the ketchup bottles to this day. Heinz seems to still be attached to the number 57: for example, its 1934 cookbook had 57 recipes, and its headquarters are P.O. Box 57. Heinz 57 is even sometimes used as slang for a dog with mixed breeds, or a person from mixed ethnicities.Why was 57 chosen? Heinz said that he chose 57 because 5 was his wife's lucky number and 7 was his, and also because the number ends in 7, and numbers that end in 7 tend to be more appealing than other numbers. In his own words, Heinz said that seven was chosen largely because of "the psychological influence of that figure and of its enduring significance to people of all ages". (I'm telling you, numbers ending in 7 really do appear disproportionately often).Fifty-eight5858 = 2+3+5+7+11+13+17, the sum of the first 7 primes, and one less than the 17th prime number, 59. This means that 58 has connections with my favorite numbers 7 and 17. The first few numbers that are the sum of the first x primes are 2, 5, 10, 17, 28, 41, 58, 77, 100, 129, 160...Sums of primes seem to often have special properties - the case of 58 isn't too interesting, but there are some cool examples, like 39, 77, and 100.Fifty-nine59The seventeenth prime number (sorry I had to)Also, 59 is an example of a weak factorial prime, a type of prime number I coined in analogy to factorial primes and primorial primes. A weak factorial prime is a prime number one more or less than a weak factorial - the sequence starts with 2, 3, 5, 7, 11, 13, 59, 61, 419, 421, 839, 2521, but the next isn't until 232,792,559. It's interesting to note that all numbers neighboring a weak factorial (other than 1) are prime up to a certain point, but soon after, weak factorial primes become pretty sparse. See also 841.59 is also the number of stellations of an icosahedron, counting the icosahedron itself as a stellation.Sixty60Sixty is a highly composite number, a number with more divisors than any smaller number.60 is the "weak factorial of five", meaning it's the smallest number divisible by 1 through 5. It's also the weak factorial of 6 since 60, the smallest number divisible by 1 through 5, is already divisible by 6. See 420 for more.60 was used as a base by the Babylonians, with a fairly intricate numeral system with 3, 10, and 30 all used as sub-bases. See my large number timeline for more.Because of that usage, there are 60 seconds in a minute, and 60 minutes in an hour, making 60 important in everyday life. Sixty was a good choice because it is divisible by many numbers: 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 can all be divided into 60.Sixty-one / gagthree61Sixty-one is a weak factorial prime (see 59) and twin primes with 59, centered around the important number 60.More notable to googology, 61 is expressible as A(3,3) in the best known version of the Ackermann function. The Ackermann function is defined as follows:A(a,b) =b+1 if a = 0A(a-1,1) if b = 0A(a-1,A(a,b-1)) otherwiseSo A(3,3) begins solving as follows:A(3,3)= A(2,A(3,2))= A(2,A(2,A(3,1)))= A(2,A(2,A(2,A(3,0))))= A(2,A(2,A(2,A(2,1))))= A(2,A(2,A(2,A(1,A(2,0)))))= A(2,A(2,A(2,A(1,A(1,1)))))= A(2,A(2,A(2,A(1,A(0,A(1,0))))))= A(2,A(2,A(2,A(1,A(0,A(0,1))))))= A(2,A(2,A(2,A(1,A(0,2)))))= A(2,A(2,A(2,A(1,3))))= A(2,A(2,A(2,A(0,A(1,2)))))= A(2,A(2,A(2,A(0,A(0,A(1,1))))))= A(2,A(2,A(2,A(0,A(0,A(0,A(1,0)))))))= A(2,A(2,A(2,A(0,A(0,A(0,2))))))= A(2,A(2,A(2,A(0,A(0,3)))))= A(2,A(2,A(2,A(0,4))))= A(2,A(2,A(2,5))))= A(2,A(2,A(1,A(2,4)))))...and with continued iteration and iteration and iteration we eventually get to ... 61.If it takes that long just to get 61, then how are we supposed to even estimate higher values? Fortunately, there are actually easy mathematical formulas to make computing the Ackermann function much faster:A(0,n) = n+1A(1,n) = n+2A(2,n) = 2n+3A(3,n) = 2^(n+3)-3A(4,n) = 2^^(n+3)-3A(5,n) = 2^^^(n+3)-3etc.This makes computing A(3,3) a lot less of a hassle.The Ackermann function has a surprisingly innocent definition, and A(3,3) is not too bad of a number, but starting with A(4,x) it QUICKLY blasts into the stars - see my article on the function for more.61 is a number expressible with the gag- prefix: gag(x) = A(x,x). Gagzero = 1, gagone = 3, gagtwo = 7, gagthree = 61, but gagfour, the next member of the sequence, is a FAR bigger number.Goobol{10,100(1)2}A goobol is a linear array of 100 tens. It's the next of Bowers' googol extensions, and the smallest of his dimensional array googolisms (though this can still be written out using linear arrays). It's a good starting point for dimensional arrays.GodgahlahE100#^#100This number is the end of Hyper-E and the beginning of Cascading-E, and was created before Cascading-E was invented. It can be written E100#^#100 in Cascading-E, and its name has a sense of far-reaching glory, unlike the goobol, a comparable googolism.Gibbol{10,100,2(1)2}Stage 1 is 10. Stage 2 is {10,10,10,10,10,10,10,10,10,10} (iteral). Stage 3 is {10,10,10,10,10.......,10,10,10} with stage 2 10s. Keep going and a gibbol is stage 100. This still isn't much further recursion from a goobol thoughGrandgahlahE100#^#100#100Also known as 99-ex-grand godgahlah, and comparable to gibbol. It’s the first new number defined with Cascading-E, as the previous godgahlah-based numbers were already created before Cascading-E.Gabbol{10,100,3(1)2}If you go back to the gibbol stages, this number is stage stage stage ... ... stage 1, with 100 stages.GreagahlahE100#^#100#100#100A bit better than grandgahlah and comparable to gabbol, but still recycling old operators. If we want to go anywhere we’ll need to define new operators. This is 99-ex-grand godgahlah-ex-grand godgahlah-ex-grand ... ... godgahlah with 100 godgahlahs.Geebol{10,100,4(1)2}
GigangahlahE100#^#100#100#100#100
Gibol{10,100,5(1)2}
GorgegahlahE100#^#100#100#100#100#100
Gobbol{10,100,6(1)2}
GulgahlahE100#^#100#100#100#100#100#100
Gabol{10,100,7(1)2}
GaspgahlahE100#^#100#100#100#100#100#100#100
Goubol{10,100,8(1)2}
GinorgahlahE100#^#100#100#100#100#100#100#100#100
Gubbol{10,100,9(1)2}
GaspgahlahE100#^#100#100#100#100#100#100#100#100#100
Gurbol{10,100,10(1)2}
GoogongahlahE100#^#100#100#100#100#100#100#100#100#100
Gebbol{10,100,11(1)2}
Goibol{10,100,12(1)2}
Guubol{10,100,13(1)2}
Geerbol{10,100,14(1)2}
Boobol{10,10,100(1)2}
GugoldgahlahE100#^#100##100Comparable to a boobol.ThroogahlahE100#^#100###100
Troobol{10,10,10,100(1)2}
ThrangahlahE100#^#100###100#100
ThrootrigahlahE100#^#100###100###100
Tribbol{10,10,10,100,2(1)2}
Gootrol{10,100(1)3}Quite a bit better than the previous attempts, this number is to goobol as goobol is to googol.Gitrol{10,100,2(1)2}Stage 1 is 10. Stage 2 is {10,10,10,10,10,10,10,10,10,10} (iteral). Stage 3 is {10,10,10,10,10.......,10,10,10} with stage 2 10s. Keep going and a gibbol is stage 100. This still isn't much further recursion from a goobol thoughGrandgahlahE100#^#100#100Also known as 99-ex-grand godgahlah, and comparable to gibbol. It’s the first new number defined with Cascading-E, as the previous godgahlah-based numbers were already created before Cascading-E.Gatrol{10,100,3(1)2}If you go back to the gibbol stages, this number is stage stage stage ... ... stage 1, with 100 stages.GreagahlahE100#^#100#100#100A bit better than grandgahlah and comparable to gabbol, but still recycling old operators. If we want to go anywhere we’ll need to define new operators. This is 99-ex-grand godgahlah-ex-grand godgahlah-ex-grand ... ... godgahlah with 100 godgahlahs.Geetrol{10,100,4(1)2}
GigangahlahE100#^#100#100#100#100
Gietrol{10,100,5(1)2}
GorgegahlahE100#^#100#100#100#100#100
Gotrol{10,100,6(1)2}
GulgahlahE100#^#100#100#100#100#100#100
Gaitrol{10,100,7(1)2}Here are some more numbers:GoutrolGutrolGurtrolGetrolGoitrolGuutrolGeertrolGaspgahlahE100#^#100#100#100#100#100#100#100
Gooquadrol{10,100(1)4}
Giquadrol{10,100,2(1)4}
Gaquadrol{10,100,3(1)4}
Geequadrol{10,100,4(1)4}
Giequadrol{10,100,5(1)4}
Goquadrol{10,100,6(1)4}
Gaiquadrol{10,100,7(1)4}Here are some more numbers:GouquadrolGuquadrolGurquadrolGequadrolGoiquadrolGuuquadrolGeerquadrolBooquadrol{10,10,100(1)4}
Biquadrol{10,10,100,2(1)4}
Baquadrol{10,10,100,3(1)4}
Beequadrol{10,10,100,4(1)4}
Trooquadrol{10,10,10,100(1)4}
Quadrooquadrol{10,10,10,10,100(1)4}
Quadreequadrol{10,10,10,10,100,4(1)4}
Quintooquadrol{10,10,10,10,10,100(1)4}
Gooquintol{10,100(1)5}
Goosextol{10,100(1)6}
Gooseptol{10,100(1)7}We can coin a series of -quadrol numbers, as well as -octol, -nonol, -decol, etc.

Google Sites

Report abuse

Page details

Page updated

Google Sites

Report abuse